Extensions to stabilised schemes

Abstract

For applying a finite element method to a given problem, it has to be written in a variational setting, which is accessible for a Galerkin scheme. Additional constraints, as for example the incompressibility condition in the Stokes problem, have to be considered as restrictions. These can be treated by the Lagrangian formalism yielding saddle point problems. One important application of such mixed systems and the corresponding finite element schemes is the following: For higher order problems one defines auxiliary variables for the derivatives. The equations describing these new quantities are handled as restrictions in the reformulated problem. In this way certain derivatives represented by auxiliary variables are approximated with higher accuracy. On the other hand the new formulation often allows to weaken the regularity assumptions on the primary solution. As one example we mention problems in perfect plasticity. Here, discontinuities may occur in the displacement field u due to slip lines in the micro-structure, whereas the stresses σ, determined by linear combinations of derivatives of u, have a smoother behaviour (see Suquet [65], Seregin [60]). From a practical point of view, the stresses are required with high accuracy. Therefore mixed methods treating the stresses directly are often more adequate for solving problems in continuum mechanics.